A191898 Symmetric square array read by antidiagonals: T(n,1)=1, T(1,k)=1, T(n,k) = -Sum_{i=1..k-1} T(n-i,k) for n >= k, -Sum_{i=1..n-1} T(k-i,n) for n < k.
1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, -1, -2, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, 1, -2, 1, 1, -2, 1, 1, 1, -1, 1, -1, -4, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, -6, -1, 1, -1, 1, -1, 1
Offset: 1
Examples
Array starts: n\k | 1 2 3 4 5 6 7 8 9 10 ----+----------------------------------------------------- 1 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 2 | 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, ... 3 | 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, ... 4 | 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, ... 5 | 1, 1, 1, 1, -4, 1, 1, 1, 1, -4, ... 6 | 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, ... 7 | 1, 1, 1, 1, 1, 1, -6, 1, 1, 1, ... 8 | 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, ... 9 | 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, ... 10 | 1, -1, 1, -1, -4, -1, 1, -1, 1, 4, ...
Links
- Antti Karttunen, Table of n, a(n) for n = 1..22155; the first 210 antidiagonals of the array
- Mats Granvik, Is this similarity to the Fourier transform of the von Mangoldt function real?
- Mats Granvik, Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?
- Mats Granvik, Primes approximated by eigenvalues?
- Mats Granvik, Are the primes found as a subset in this sequence a(n)?
- Mats Granvik, Will every eigenvalue in this type of matrix eventually be a common eigenvalue to infinitely many subsequent larger matrices of the same form?
- Mats Granvik, How write Dirichlet character sums for the terms of the von Mangoldt function?
- Mats Granvik, Do these series converge to the von Mangoldt function?
- Mats Granvik, Is this sum equal to the Möbius function?
- Mats Granvik, Can the Riemann hypothesis be relaxed to say that this matrix A consists of square roots?
- Mats Granvik, Elementary proof of the prime number theorem?
- Mats Granvik, Is this Dirichlet series generating function of the von Mangoldt function matrix correct?
- Mats Granvik, Question about ratios of polynomials evaluated at x=1
Crossrefs
Programs
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Mathematica
T[ n_, k_] := T[ n, k] = Which[ n < 1 || k < 1, 0, n == 1 || k == 1, 1, k > n, T[k, n], n > k,T[k, Mod[n, k, 1]], True, -Sum[ T[n, i], {i, n - 1}]]; (* Michael Somos, Jul 18 2011 *) (* Conjectured expression for the matrix as Dirichlet characters *) s = RandomInteger[{1, 3}]; c = RandomInteger[{1, 3}]; nn = 12; b = Table[Exp[MangoldtLambda[Divisors[n]]]^-MoebiusMu[Divisors[n]], {n, 1, nn^Max[s, c]}]; j = 1; MatrixForm[Table[Table[Product[(b[[n^s]][[m]]*DirichletCharacter[b[[n^s]][[m]], j, k^c] - (b[[n^s]][[m]] - 1)), {m, 1, Length[Divisors[n]]}], {n, 1, nn}], {k, 1, nn}]] (* Mats Granvik, Nov 23 2013 and Aug 09 2016 *) -
PARI
{T(n, k) = if( n<1 || k<1, 0, n==1 || k==1, 1, k>n, T(k, n), k -
Python
from sympy.core.cache import cacheit @cacheit def T(n, k): return 0 if n<1 or k<1 else 1 if n==1 or k==1 else T(k, n) if k>n else T(k, (n - 1)%k + 1) if n>k else -sum([T(n, i) for i in range(1, n)]) for n in range(1, 21): print([T(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Oct 23 2017
Formula
T(n,1)=1, T(1,k)=1, n>=k: -Sum_{i=1..k-1} T(n-i,k), n
T(n, n) = A023900(n). - Michael Somos, Jul 18 2011
T(n, k) = A023900(gcd(n,k)). - Mats Granvik, Jun 18 2012
Dirichlet generating function for sequence in the n-th row: zeta(s)*Sum_{ d divides n } mu(d)/d^(s-1). - Mats Granvik, Jun 18 2012 & Jun 19 2016
From Mats Granvik, Jun 19 2016: (Start)
Dirichlet generating function for the whole matrix: Sum_{k>=1} (Sum_{n>=1} T(n,k)/(n^c*k^s)) = Sum_{n>=1} (zeta(s)*Sum_{ d divides n } mu(d)/d^(s-1))/n^c = zeta(s)*zeta(c)/zeta( c + s - 1 ).
T(n,k) = A127093(n,k)^(1/2-i*a(k))*transpose(A008683(k)*(A127093(n,k)^(1/2+i*a(n)))) where a(x) is some real number. An example would be T(n,k) = A127093(n,k)^(zetazero(k))*transpose(A008683(k)*(A127093(n,k)^(zetazero(-k)))) but this is of course not special for only the zeta zeros.
Recurrence for a subset of A191898 that is a cross-directional variant of the recurrence in A051731: T(1,1)=1, T(1,2..k)=0, T(2..n,1)=0, n >= k: -Sum_{i=1..k-1} T(n-i,k) - T(n-i,k-1), n < k: -Sum_{i=1..n-1} T(k-i,n) - T(k-i,n-1). Notice that the identity matrix in linear algebra satisfies a similar recurrence:
T(1,1)=1, T(1,2..k)=0, T(2..n,1)=0, n >= k: -Sum_{i=1..n-1} T(n-i,k) - T(n-i,k-1), n < k: -Sum_{i=1..k-1} T(k-i,n) - T(k-i,n-1).
(End)
Dirichlet generating function for absolute values: Sum_{k>=1} (Sum_{n>=1} abs(T(n,k))/(n^c*k^s)) = zeta(s)*zeta(c)*zeta(s + c - 1)/zeta(2*(s + c - 1))*Product_{k>=1} (1 - 2/(prime(k) + prime(k)^(s + c))). After Vaclav Kotesovec in A173557. - Mats Granvik, Apr 25 2021
A199515 Denominators of: MoebiusMu(n)^2*(n/(n - EulerPhi(n))).
1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 4, 7, 1, 1, 1, 1, 1, 3, 6, 1, 1, 1, 7, 1, 1, 1, 11, 1, 1, 13, 9, 11, 1, 1, 10, 5, 1, 1, 5, 1, 1, 1, 12, 1, 1, 1, 1, 19, 1, 1, 1, 3, 1, 7, 15, 1, 1, 1, 16, 1, 1, 17, 23, 1, 1, 25, 23, 1, 1, 1, 19, 1, 1, 17, 9, 1, 1, 1, 21, 1, 1, 21, 22, 31, 1, 1, 1
Offset: 2
Comments
Denominators of zeros to a symmetric polynomial.
See A199514 for numerators and program.
Links
- Antti Karttunen, Table of n, a(n) for n = 2..65537
Programs
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Mathematica
Table[MoebiusMu[n]^2 (n/(n-EulerPhi[n])),{n,2,90}]//Denominator (* Harvey P. Dale, May 15 2022 *) -
PARI
A199515(n) = denominator(issquarefree(n)*(n/(n-eulerphi(n)))); \\ Antti Karttunen, Sep 07 2018
A231425 The Schramm triangle: T(n,k) = f(gcd(n,k)), where f = Dirichlet inverse of Euler totient.
1, 1, -1, 1, 1, -2, 1, -1, 1, -1, 1, 1, 1, 1, -4, 1, -1, -2, -1, 1, 2, 1, 1, 1, 1, 1, 1, -6, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, -1, 1, -1, -4, -1, 1, -1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -10, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2
Offset: 1
Comments
When taking matrix powers of the reversed triangle it might be more interesting to start with the first term T(1,1) set to 0.
Three fundamental number theoretic sequences are found from this triangle. The first is the Dirichlet inverse of the Euler totient which are the entries themselves. The Greatest Common Divisor-Fourier transform described by Wolfgang Schramm gives the Möbius function times n =1, -2, -3, 0, ... = A055615, as follows:
1*cos(-2*k*Pi/n) = 1
1*cos(-2*k*Pi/n) -1*cos(-2*k*Pi/n) = -2
1*cos(-2*k*Pi/n) +1*cos(-2*k*Pi/n) -2*cos(-2*k*Pi/n) = -3
The two components in this GCD-Fourier triangle both sum to the sequence 1,0,0,0,0, ... A000007.
1 = 1
1 -1 = 0
1 +1 -2 = 0
...
cos(-2*k*Pi/n) = 1
cos(-2*k*Pi/n), cos(-2*k*Pi/n) = 0
cos(-2*k*Pi/n), cos(-2*k*Pi/n), cos(-2*k*Pi/n) = 0
...
This latter Fourier transform like triangle is also called the chaotic set by some authors.
The third arithmetic sequence is the von Mangoldt function that can be computed as sums with periods equal to rows in this triangle:
1
log(2) = Sum_{n>=0} (1/(n+1) -1/(n+2))
log(3) = Sum_{n>=0} (1/(n+1) +1/(n+2) -2/(n+3))
log(2) = Sum_{n>=0} (1/(n+1) -1/(n+2) +1/(n+3) -1/(n+4))
log(5) = Sum_{n>=0} (1/(n+1) +1/(n+2) +1/(n+3) +1/(n+4) -4/(n+5))
log(1) = Sum_{n>=0} (1/(n+1) -1/(n+2) -2/(n+3) -1/(n+4) +1/(n+5) +2/(n+6))
...
Also the matrix inverse of the reversal of this number triangle gives the all-ones sequence in the first column. Therefore this number triangle is a companion to A054524.
Links
- G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Programs
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Mathematica
Clear[nn, t, n, k]; nn = 12; t[n_, 1] = 1; t[1, k_] = 1; t[n_, k_] := t[n, k] = If[n >= k, -Sum[t[n - i, k], {i, 1, k - 1}], -Sum[t[k - i, n], {i, 1, n - 1}]]; Flatten[Table[Table[t[n, k], {k, 1, n}], {n, 1, nn}]]
Formula
T(n,k) = A023900(gcd(n,k)) for n >= k.
Comments